scholarly journals Bass series of local ring homomorphisms of finite flat dimension

1993 ◽  
Vol 335 (2) ◽  
pp. 497-523 ◽  
Author(s):  
Luchezar L. Avramov ◽  
Hans-Bjørn Foxby ◽  
Jack Lescot
Keyword(s):  
Local Ring ◽  
Ring Homomorphisms ◽  
Flat Dimension

scholarly journals Ascent properties for test modules

2021 ◽  
pp. 153-167
Author(s):  
Keri Sather-Wagstaff
Keyword(s):  
Local Ring ◽  
Projective Dimension ◽  
Graded Algebras ◽  
Ring Homomorphisms ◽  
Flat Dimension ◽  
Finite Dimensional ◽  
Differential Graded Algebras ◽  
Homological Dimensions ◽  
Tor Modules

We investigate modules for which vanishing of Tor-modules implies finiteness of homological dimensions (e.g., projective dimension and G-dimension). In particular, we answer a question of O. Celikbas and Sather-Wagstaff about ascent properties of such modules over residually algebraic flat local ring homomorphisms. To accomplish this, we consider ascent and descent properties over local ring homomorphisms of finite flat dimension, and for flat extensions of finite dimensional differential graded algebras.

Complete Intersection Flat Dimension and the Intersection Theorem

2012 ◽  
Vol 19 (spec01) ◽  
pp. 1161-1166
Author(s):  
Parviz Sahandi ◽  
Tirdad Sharif ◽  
Siamak Yassemi
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Intersection Theorem ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Gorenstein Dimension ◽  
Flat Dimension ◽  
Gorenstein Flat ◽  
Similar Extension

Any finitely generated module M over a local ring R is endowed with a complete intersection dimension CI-dim RM and a Gorenstein dimension G-dim RM. The Gorenstein dimension can be extended to all modules over the ring R. This paper presents a similar extension for the complete intersection dimension, and mentions the relation between this dimension and the Gorenstein flat dimension. In addition, we show that in the intersection theorem, the flat dimension can be replaced by the complete intersection flat dimension.

Restricted injective dimensions over local homomorphisms

2018 ◽  
Vol 68 (3) ◽  
pp. 691-697 ◽  
Author(s):  
Dejun Wu ◽  
Fangdi Kong
Keyword(s):  
Local Ring ◽  
Injective Dimension ◽  
Ring Homomorphisms

Abstract In this paper, we study the small restricted injective dimension over local ring homomorphisms. Some of known results are generalized. For example, the Bass formula for the small restricted injective dimension of complexes is extended.

scholarly journals Lower bounds for the number of semidualizing complexes over a local ring

2012 ◽  
Vol 110 (1) ◽  
pp. 5 ◽  
Author(s):  
Sean Sather-Wagstaff
Keyword(s):  
Local Ring ◽  
Lower Bounds ◽  
Ring Homomorphism ◽  
Order Structure ◽  
Noetherian Local Ring ◽  
Isomorphism Classes ◽  
Flat Dimension ◽  
A Chain ◽  
Dualizing Complexes

We investigate the set $\mathfrak(R)$ of shift-isomorphism classes of semi-dualizing $R$-complexes, ordered via the reflexivity relation, where $R$ is a commutative noetherian local ring. Specifically, we study the question of whether $\mathfrak(R)$ has cardinality $2^n$ for some $n$. We show that, if there is a chain of length $n$ in $\mathfrak(R)$ and if the reflexivity ordering on $\mathfrak (R)$ is transitive, then $\mathfrak(R)$ has cardinality at least $2^n$, and we explicitly describe some of its order-structure. We also show that, given a local ring homomorphism $\varphi\colon R\to S$ of finite flat dimension, if $R$ and $S$ admit dualizing complexes and if $\varphi$ is not Gorenstein, then the cardinality of $\mathfrak (S)$ is at least twice the cardinality of $\mathfrak (R)$.

scholarly journals Reflexivity and Ring Homomorphisms of Finite Flat Dimension

2007 ◽  
Vol 35 (2) ◽  
pp. 461-500 ◽  
Author(s):  
Anders Frankild ◽  
Sean Sather-Wagstaff
Keyword(s):  
Ring Homomorphisms ◽  
Flat Dimension

scholarly journals Ascent Properties of Auslander Categories

2009 ◽  
Vol 61 (1) ◽  
pp. 76-108 ◽  
Author(s):  
Lars Winther Christensen ◽  
Henrik Holm
Keyword(s):  
Recent Work ◽  
Local Ring ◽  
Homomorphic Image ◽  
Base Change ◽  
Flat Dimension ◽  
Homological Dimensions ◽  
Gorenstein Local Ring

Abstract. Let R be a homomorphic image of a Gorenstein local ring. Recent work has shown that there is a bridge between Auslander categories and modules of finite Gorenstein homological dimensions over R. We use Gorenstein dimensions to prove new results about Auslander categories and vice versa. For example, we establish base change relations between the Auslander categories of the source and target rings of a homomorphism φ: R → S of finite flat dimension.

On Generalized Regular Local Ring

2015 ◽  
Vol 3 (1) ◽  
pp. 145-152
Author(s):  
Zubayda Ibraheem ◽  
Naeema Shereef
Keyword(s):  
Local Ring ◽  
Regular Local Ring

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

On equimultiplicity

1982 ◽  
Vol 91 (2) ◽  
pp. 207-213 ◽  
Author(s):  
M. Herrmann ◽  
U. Orbanz
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Analytic Spread

This note consists of some investigations about the condition ht(A) = l(A) where A is an ideal in a local ring and l(A) is the analytic spread of A (9).In (4) we proved the following: If R is a local ring and P a prime ideal such that R/P is regular then (under some technical assumptions) ht(P) = l(P) is equivalent to the equimultiplicity e(R) = e(RP). Also for a general ideal A (which need not be prime), the condition ht(A) = l(A) can be translated into an equality of certain multiplicities (see Theorem 0).

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